Page 53 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH02_52_08/30/05

Basic Members: Lumped- and Distributed-Parameter Modeling and Design

52 Chapter Two

2
l d f (x) 2
= E
k l,y ฒ I (x) l dx
y
0 dx
2
l d f (x) 2
r
k r,y ฒ I (x) dx dx (2.19)
= E
y
0
2
2
l d f (x) d f (x)
k c,y ฒ I (x) l r dx
= E
y
0 dx dx
The bending about the other axis, the z axis, can be treated similarly,
and the equations that have been presented for the bending about
the y axis are valid for the z axis by interchanging of the y and z
subscripts in Eqs. (2.14) through (2.19). The stiffness matrix collecting
axial, torsion, and two-axis bending effects can be expressed as
k k 0 0 0 0
l,y c,y
k c,y k r,y 0 0 0 0
0 0 k l,z k c,z 0 0
K = (2.20)
0 0 k c,z k r,z 0 0

0 0 0 0 k a 0
0 0 0 0 0 k t

and a load deformation equation can be expressed in matrix form as
{L} = K {d} (2.21)

where the load vector is
{L} = {F M F M F M } T (2.22)
y
y
x
x
z
z
and the deformation/displacement vector is
{d} = {u ș u ș u ș } T (2.23)
x x
y z
y
z
Compliance approach. The second approach to finding the relevant
stiffnesses of fixed-free straight line members addresses the compliance
(or flexibility) method, which utilizes Castigliano’s second (or displace-
ment) theorem to find the relationship between an elastic deformation

Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com)
Copyright © 2004 The McGraw-Hill Companies. All rights reserved.
Any use is subject to the Terms of Use as given at the website.